Nonlinear Goal-Oriented Bayesian Inference: Application to Carbon Capture and Storage

In many engineering problems, unknown parameters of a model are inferred in order to make predictions, to design controllers, or to optimize the model. When parameters are distributed (continuous) or very high-dimensional (discrete) and quantities of interest are low-dimensional, parameters need not be fully resolved to make accurate estimates of quantities of interest. In this work, we extend goal-oriented inference—the process of estimating predictions from observed data without resolving the parameter, previously justified theoretically in the linear setting—to Bayesian statistical inference problem formulations with nonlinear experimental and prediction processes. We propose to learn the joint density of data and predictions offline using Gaussian mixture models. When data are observed online, we condition the representation to arrive at a probabilistic description of predictions given observed data. Our approach enables real-time estimation of uncertainty in quantities of interest and renders tractable high-dimensional PDE-constrained Bayesian inference when there exist low-dimensional output quantities of interest. We demonstrate the method on a realistic problem in carbon capture and storage for which existing methods of Bayesian parameter estimation are intractable.